CN-TST 2016 Problem 17

Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$ , $\Omega$ , is a $2016-$ gon $A_1A_2\ldots A_{2016}$ . Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$ , such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other. Draw triangles whose vertices are in $S$ , such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$ . Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.